4. Modeling and simulation
4.4 Data analysis
4.4 Data analysis
During the simulation the time dependence of the fluorescence intensity distribution was recorded in the same cross section of the nucleus as in the FRAP experiment. The experimental and simulated images were compared by their cross correlation coefficient,
𝑐𝑘,𝑙 = 1
𝑁𝜎𝑘𝜎𝑙 𝑣𝑘 𝑥, 𝑦 − 𝑣𝑘
𝑥.𝑦
𝑣𝑙 𝑥, 𝑦 − 𝑣𝑙 , (10)
where 𝑣𝑘 𝑥, 𝑦 is the pixel intensity of the image, N is the number of pixels, 𝑣𝑘 is their average intensity, 𝜎𝑘 is their standard deviation, subscripts k and l refer to the two series of images.
Cross correlation results were improved by removing the background from all the images, and a mask was used to restrict the region analyzed. These manipulations reduced perturbing effects caused by motion and deformation of the cell.
The different liquid phases of the cell were described by three relaxation times, one for the cytosol τcyt , one for the nucleosol τnsol and one for the effective substance of the nuclear envelope τne . Simulation time step δt was a fitting parameter. For each experimental image k there was a global maximum lmax(k) in the cross-correlation coefficient, as shown in Figure 9.
31
Figure 9. Cross-correlation coefficient 𝑐𝑘,𝑙 (red, blue and green lines) of the 2d, 5th and 10th frames of measured and simulated FRAP date. The cross-correlation crosses denote
their global maxima.
When the global maximum was a linear function of k, the real and digital cells were assume to correspond to each other. The values of τcyt and τne were varied so as to maximize the linearity of lmax(k), which gave the simulation time as a function of real (experimental) time.
32 5. Results
5.1 The Axelrod/Soumpasis method
FRAP experiments were done on EYFP and H2B-ECFP-expressing HeLa cells. In these experiments 10 cells were measured and first analyzed by the method of Axelrod/Soumpasis. According to this analysis, the average diffusion coefficient of the measured cells was 3.1 ±1.1 μm2/s. As can be seen from Figure 10, the model used did not fit the data well. The reason for this discrepancy is that the assumptions of the Axelrod/Soumpasis method were not really satisfied in the experiment, which caused the low value of the diffusion coefficient in the nucleus in comparison with more accurate values reported in [Kuhn 2011]. A curve fitted to a set of recovery data by the free diffusion model of Soumpasis is shown in Figure 10.
Figure 10. A set of measured recovery data with Axelrod normalization and a fit of that set by the free diffusion model of Soumpasis.
33
5.2 Results of FRAP experiments and simulations
With the new methods which assumed that transport was that through a porous medium allowed us to determine the diffusion coefficients in the nucleosol, cytosol and nuclear envelope shown in Table 1.
Table 1. The diffusion coefficients as found by LBM for the nucleosol, cytosol and nuclear envelope of HeLa cells, their average values and standard deviations (STD).
HeLa Dnuc [µm2/s] Denv [µm2/s] Dcyt [µm2/s] and comparing the resulting fluorescence distribution with that of the corresponding numerical simulation. The new approach gave excellent linear correlation between the
34
frames of the experiments and the related simulations, an example of which is shown in Figure 11. The resulting nucleosol diffusion coefficient, Dnuc, was 29.2 ±4.5 μm2/s. The method also produced diffusion coefficients for the cytosol and nuclear envelope, although the main organelle explored was the nucleus. The former values were not determined accurately. Nevertheless, the cytosol diffusion coefficient, Dcyt, was found to be 30.9 ±10.8 μm2/s and that of the nuclear envelope, Denv, was 0.2 ±0.2 μm2/s.
Figure 11. Correspondence between highest cross correlation values of an experiment and the corresponding simulation.
35 6. Discussion
Modeling involves developing a physical, conceptual and computer-based representation of the system considered. In this work a model which enabled analysis of nucleocytoplasmic diffusion of proteins in living HeLa cells was represented.
Experiments were made in live cells with fluorescent proteins, confocal microscopy was used to acquire 3D data and ImageJ software was used to perform image analysis.
The results obtained in this way by FRAP showed that EYFP is freely diffusing inside the nucleus, which was demonstrated by the rapid recovery rate of free EYFP (Figure 8). On the other hand a conventional FRAP analysis produced very low diffusion coefficients, which means that the conditions in the measurements did not correspond to those assumed in the analysis.
A fully numerical modeling approach applied to diffusion was the LB method. The heterogeneous fluorescence intensity in the nucleoplasm was interpreted as a homogeneous distribution in the nucleosol, the liquid phase of the nucleoplasm. The plasma membrane was represented as an impermeable boundary and the nuclear envelope was considered as a permeable layer with a diffusion coefficient of its own. The method was not fine-tuned however so as to be able to determine τcyt and τne very accurately, and that is why their values varied quite much.
The single colour fluorescence correlation spectroscopy (scFCS) technique in living cells has been used to show that the diffusion coefficient (D) of the fast fraction of EGFP molecules is 23.0 ± 1.0 µm2/s (SEM) and 25.1 ± 1.1 µm2/s in the nucleus and in the cytoplasm respectively [Maertens 2005]. These results are well comparable with the diffusion coefficients found here by the LB method.
There are a few possible sources of error in the present method. It is important that the nuclear envelope and the nucleus are reliably indentified. For that purpose the histone fusion protein was used to label the chromatin. Cells had a tendency to move during measurements, which had to be taken into account in the correlation analysis. Also, the
36
cross section analyzed at the imaging phase had to be indentified properly. Non-specific binding of the protein was not taken into account.
For clarity a freely diffusive and non-binding protein was selected. Binding reactions specific for the nucleus will obviously affect the protein mobility. Within the present methodology, binding/dissociation reactions with protein receptors can as well be taken into account.
37 7. Conclusions
Analysis of fluorescence recovery after photobleaching can be used to determine the dynamic parameters of proteins, including their diffusion coefficients, mobile fractions, transport rates and binding/dissociation rates. Here we focused on their diffusion coefficients.
A numerical model for FRAP was used to determine the diffusion properties of proteins by including the effect of the plasma membrane, the nuclear envelope, the cell nucleus, the fibrous structures of the cytoplasm and the chromatin, which reduced protein mobility. As this method simulated the fluorescence distribution in the entire cell, there was no need to make additional assumptions about the bleach process, such as e.g. the shape of the laser profile or its duration. The present method removed the difficulties of the conventional analysis, could produce, interesting results for protein diffusion in the cell as demonstrated above and can be applied in the future to define protein interactions beyond pure diffusion.
38 Appendix 1
Materials and methods
1. Cell culture
HeLa cells were used in this research. Cells were maintained in Dulbecco‟s Modified Eagle Medium (DMEM, Gibco Introgen, Paisley, UK). They were grown as monolayer in 75 cm2 culture flasks (Sarstedt Inc., Newton, USA) maintained in a 5% CO2 incubator at +37 ºC. Cells were passaged twice a week.
2. Transfection
EYFP and H2B-ECFP constructs were transfected to HeLa cells with the TransIT Transfection Reagent (Mirus). Distributions of proteins are shown in Figure 12. Cells were growing on 50 mm culture dishes for live imaging. The protocol of cell transfection on 50mm culture dishes is described below. 15 µl of TransIT reagent was added to 650 µl of serum-free DMEM and incubated for 15 minutes. 4 µl of plasmid DNA was added to the transfection solution, and incubation at RT was continued for half an hour. The medium on the culture dish was replaced with fresh DMEM, and the transfection solution was added into the dish, followed by incubation overnight at +37 ºC.
39
A B
Figure 12. Images of HeLa cells showing the distributions of fluorescent proteins. A.
Distribution of EYFP which is a small, noninteracting protein that diffuses freely in the entire cell. EYFP is very good for photobleaching experiments. B. H2B ECFP shows the
location and relative concentration of chromatin inside the nucleus and allows thus to deduce the position of the nucleus quite accurately.
3. Live cell imaging with confocal microscope
Measurements were done with an Olympus confocal microscope. Object was UPLSAPO 60x (numerical aperture: 1.20). A specific region was bleached and then the sample was scanned to find how fast (in frames) the bleached region was recovered. A selected cell was bleached with the “tornado” method in a region of circular cross section of 10x10 pixels with 515 nm laser wavelength and 100 % laser power. Every sample was bleached only once. Bleaching was started after 10 frames were taken of the cell, and after the bleach additional 100 frames were recorded. Scanning was done with the maximum scanning speed. 12 nuclei were subjected to a FRAP experiment, and the data obtained were used to determine the diffusion coefficient of pEYFP-N3 in the imaged cells.
40
The bleached regions were chosen so that they were not near any cell organelle or in the nucleus near nucleolus because then the recovery would have taken longer and the results would have been less reliable.
4. Image analysis
The data obtained from the confocal microscope were analyzed with an image analysis program, in this case the ImageJ software. This kind of image analysis program makes the analysis of microscopy images straightforward and easy to perform.
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